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Journal of Convex Analysis 15 (2008), No. 1, 105--129
Copyright Heldermann Verlag 2008

Singularities for a Class of Non-Convex Sets and Functions, and Viscosity Solutions of some Hamilton-Jacobi Equations

Giovanni Colombo
Dip. di Matematica Pura e Applicata, UniversitÓ di Padova, Via Trieste 63, 35121 Padova, Italy

Antonio Marigonda
Dip. di Matematica "Felice Casorati", UniversitÓ di Pavia, Via Ferrata 1, 27100 Pavia, Italy


We study nondifferentiability points for a class of continuous functions $f:\mathbb R^N\to\mathbb R$ whose epigraph satisfies a kind of external sphere condition with uniform radius (called $\varphi$-convexity or proximal smoothness). The functions belonging to this class are not necessarily Lipschitz. However, they enjoy some properties analogous to semiconvex functions; in particular they are twice $\mathcal L^{N}$-a.e.\ differentiable (see the authors in Calc. Var. 25 (2006) 1--31). In partial analogy with the study of singularities of semiconcave functions (see P. Cannarsa, C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control", Birkh\"auser, Boston (2004)), under suitable conditions we give estimates from below of the nondifferentiability set, which consists of points where the subdifferential is not a singleton, as well as (differently from semiconvex functions) of points where it is empty. Furthermore, we show that if a function in this class is an a.e. solution of a Hamilton-Jacobi equation, then under suitable assumptions it is actually a viscosity solution. Methods of nonsmooth analysis and geometric measure theory are used, including a representation of Clarke's generalized gradient as the closed convex hull of limits of Fr\'echet derivatives.

Keywords: Nonsmooth analysis, proximal smoothness, semiconvex functions.

MSC: 49J52; 49L25

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